Now we try to carry out the time evolution of the scattering data. If \(q\)  also depends on time variable \(t\) , we can obtain the functions
\(a(z)\)  and \(b(z)\)  mentioned above for all times \(t\in \mathbb {R}\) . This can be seen applying \(\partial _t\)  to (REF ) and some
standard arguments can be taken, for example [1]}, [2]}. Then time dependence of scattering data which can be expressed
in terms of the following replacement
\(&c(\eta _n)\rightarrow c(t,\eta _n)=c(0, \eta _n)e^{\lambda (\eta _n)(4k^2(\eta _n)+2)t},\\&r(z)\rightarrow r(t,z)=r(0,z)e^{\lambda (4k^2+2)t}\) 
